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Wize University Calculus 1 Textbook > Applications of Differentiation

Maximum and Minimum on Closed Intervals

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Maximum and Minimum on Closed Intervals

A continuous function on a closed interval will always have a largest and smallest value. The values are called Absolute Extrema or Global Extrema.

Absolute Extrema

A function f(x)f(x) has an absolute maximum at a point xx in its domain if f(x)≥f(y) f(x)\ge f(y) for every yy in the domain.



A function f(x)f(x) has an absolute minimum at a point xx in its domain if f(x)≤f(y)f(x) \le f(y)for every yy in the domain.



Note: This definition can also be applied to a function on an interval. The absolute extrema are the biggest and smallest value achieved by the function on that interval.

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Maximum and Minimum on Closed Intervals

For a continuous function f(x)f(x) on a closed interval [a,b],[a, b], there exists an absolute minimum and an absolute maximum.

Watch Out!
This is not necessarily true on an open interval!

Procedure to find the Absolute Extrema of a function on a Closed Interval [a,b]

  1. Find all critical points and singular points of f(x)f(x) in the open interval (a,b)(a, b)
  2. Compute f(x)f(x) at the critical points and singular points and at the endpoints x=ax = a and x=bx = b
  3. The largest and smallest values found are the absolute maximum and minimum respectively
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Example: Finding Absolute Extrema

Find the absolute extrema (max/min) of f(x)=e−x2+1f\left(x\right)=e^{-x^2}+1 on the interval [−1, ∞)\left[-1,\ \infty\right).

1. Find all critical points of the function
The first derivative is f′(x)=e−x2(−2x)f'\left(x\right)=e^{-x^2}\left(-2x\right).
f′(x)=0   →   e−x2(−2x)=0   →   x=0f'\left(x\right)=0\ \ \ \to\ \ \ e^{-x^2}\left(-2x\right)=0\ \ \ \to\ \ \ x=0

2. Compute the yy value of the critical points
*If an interval is given, compute the yy values at these end points

f(0)=e−02+1=2f\left(0\right)=e^{-0^2}+1=2

f(−1)=e−(−1)2+1=e−1+1f\left(-1\right)=e^{-\left(-1\right)^2}+1=e^{-1}+1

"Sub" in the right end-point:

f(∞)→e−(∞)2+1=e−∞+1=0+1=1f\left(\infty\right)\to e^{-\left(\infty\right)^2}+1=e^{-\infty}+1=0+1=1


3. The largest and smallest values that we find are the absolute max and min, respectively
Observe that 1<e−1+1<21<e^{-1}+1<2.

Therefore, the absolute max point is (0, 2)\left(0,\ 2\right) and there is no absolute min point, but instead, as x→∞x\to\infty, there is a horizontal asymptote at y=1y=1 (all values are above this line)
Find the absolute minimum of f(x)=arcsin(x2)f(x)=\text{arcsin}\left(\frac{x}{2}\right) on the interval [1,2][1,2].
Find the absolute maximum and minimum of value of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]






Extra Practice
Maximum and Minimum on Closed Intervals
Find the absolute max of the function f(x)=cos⁡x⋅e−xf\left(x\right)=\cos x\cdot e^{-x} on [0,π]\left[0,\pi\right]
Closed Intervals: Maximum and Minimum
Find the local and absolute extreme values of f(x)=3x−4x2+1f(x)=\frac{3x-4}{x^2+1} on the interval [−2,2][-2,2]
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Concavity and Inflection Points: Graph of a Derivative
Practice: Graph of a Derivative
Suppose that ff is a differentiable function on the interval [−4,2]\left[-4,2\right]. If the following graph represents the derivative of ff, which of the following statements is correct?
Determine where the absolute extrema of f(x)=3x23−2xf(x)=3x^{\frac{2}{3}}-2x on the interval [−1,1] [-1,1] occur, and their values.
Practice: Absolute Extrema on Closed Interval
Q.\textbf{Q.} Find the absolute maximum and minimum values of f(x)=6x43−3x13\displaystyle f(x)=6x^{\frac{4}{3}}-3x^{\frac{1}{3}} on the interval [−1,1][-1,1] and then everywhere, using the second derivative test.
Practice: Relative Extrema with Trig
Q.\textbf{Q.} Find all the relative extrema of f(x)=2ex(sin⁡x−cos⁡x)\displaystyle f(x)=2e^x(\sin{x}-\cos{x}) on the interval [π2,π]\bigg[\dfrac{\pi}{2},\pi\bigg].
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Practice: Graph of a Derivative (~F2018 Final Q17) (~F2016 Final Q16)
Practice: Graph of a Derivative
Suppose that ff is a differentiable function on the interval (−4,2)(-4,2). If the following graph represents the derivative of ff, which of the following statements is correct?
Maximum and Minimum on Closed Intervals
Find the absolute max of the function f(x)=cos⁡x⋅e−xf\left(x\right)=\cos x\cdot e^{-x} on [0,π]\left[0,\pi\right]
Critical Points and Extrema: Maximum and Minimum on Closed Intervals
Find the absolute maximum of value of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1}
on the interval [0,2][0,2].
Critical points and extrema: Maximum and minimum on closed intervals
Find the absolute maximum value of f(x)=2ex(sin⁡x−cos⁡x)f(x)=2e^x(\sin x-\cos x)
on the interval [π/2,π][\pi/2,\pi].
Critical points and extrema: Maximum and Minimum on closed intervals
Find the absolute minimum of f(x)=arcsin(x2)f(x)=\text{arcsin}\left(\frac{x}{2}\right) on the interval [1,2][1,2].
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Find the absolute maximum value of f(x)=2ex(sin⁡x−cos⁡x)f(x)=2e^x(\sin x-\cos x)
on the interval [π/2,π][\pi/2,\pi].
Find the absolute minimum and absolute maximum of f(x)=x13(7−x)2 f\left(x\right)=x^{\frac{1}{3}}\left(7-x\right)^2\ on [−1,7]\left[-1,7\right].
Practice: Relative Extrema
Practice Question: Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Closed Intervals: Maximum and Minimum
Find the local and absolute extreme values of f(x)=3x−4x2+1f(x)=\frac{3x-4}{x^2+1} on the interval [−2,2][-2,2]
The absolute maximum value of f(x)=1−x23x+5f(x)=\dfrac{1-x^2}{3x+5} , over the closed interval [−1,2][-1,2] is:
final114
The absolute maximum value of f(x)=1−x23x+5f(x)=\dfrac{1-x^2}{3x+5} , over the closed interval [−1,2][-1,2] is:
final114
The absolute maximum value of f(x)=1−x23x+5f(x)=\dfrac{1-x^2}{3x+5} , over the closed interval [−1,2][-1,2] is:
Find the absolute minimum of f(x)=arcsin(x2)f(x)=\text{arcsin}\left(\frac{x}{2}\right) on the interval [1,2][1,2].
Practice: Absolute Extrema on Closed Interval
Q.\textbf{Q.} Find the absolute maximum and minimum values of f(x)=6x43−3x13\displaystyle f(x)=6x^{\frac{4}{3}}-3x^{\frac{1}{3}} on the interval [−1,1][-1,1] and then everywhere, using the second derivative test.
Practice: Relative Extrema with Trig
Q.\textbf{Q.} Find all the relative extrema of f(x)=2ex(sin⁡x−cos⁡x)\displaystyle f(x)=2e^x(\sin{x}-\cos{x}) on the interval [π2,π]\bigg[\dfrac{\pi}{2},\pi\bigg].
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Determine where the absolute extrema of f(x)=3x23−2xf(x)=3x^{\frac{2}{3}}-2x on the interval [−1,1] [-1,1] occur, and their values.
The absolute maximum value of f(x)=1−x23x+5f(x)=\dfrac{1-x^2}{3x+5} , over the closed interval [−1,2][-1,2] is:
Find the absolute minimum and absolute maximum of f(x)=x13(7−x)2  on [−1,7]f\left(x\right)=x^{\frac{1}{3}}\left(7-x\right)^2\ \text{ on }\left[-1,7\right].
For the following function, determine all the local and global minimum/maximum and inflection points over the given interval
h(x)=x5−x3+1over [−2,2]h(x)= x^5-x^3+1\quad \text{over}\ [-2,2]
For the following function, determine all the local and global minimum/maximum and inflection points over the given interval
f(x)=x3−3x2 over  [−1,3]f(x)=x^3-3x^2\quad \text{ over }\ [-1,3]
Find the absolute maximum of value of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2].
Find the absolute minimum and absolute maximum of f(x)=x13(7−x)2 f\left(x\right)=x^{\frac{1}{3}}\left(7-x\right)^2\ on [−1,7]\left[-1,7\right].
Concavity and Inflection Points: Graph of a Derivative
Practice: Graph of a Derivative
Suppose that ff is a differentiable function on the interval [−4,2]\left[-4,2\right]. If the following graph represents the derivative of ff, which of the following statements is correct?